Given a real number τ, we study the approximation of τ by signed harmonic sums σN(τ):=∑n≤Nsn(τ)/n, where the sequence of signs (sN(τ))N∈N is defined “greedily” by setting sN+1(τ):=+1 if σN(τ)≤τ, and sN+1(τ):=−1 otherwise. More precisely, we compute the limit points and the decay rate of the sequence (σN(τ)−τ)N∈N. Moreover, we give an accurate description of the behavior of the sequence of signs (sN(τ))N∈N, highlighting a surprising connection with the Thue–Morse sequence.

Greedy approximations by signed harmonic sums and the Thue–Morse sequence / Bettin, Sandro; Molteni, Giuseppe; Sanna, Carlo. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - STAMPA. - 366:(2020), p. 107068. [10.1016/j.aim.2020.107068]

### Greedy approximations by signed harmonic sums and the Thue–Morse sequence

#### Abstract

Given a real number τ, we study the approximation of τ by signed harmonic sums σN(τ):=∑n≤Nsn(τ)/n, where the sequence of signs (sN(τ))N∈N is defined “greedily” by setting sN+1(τ):=+1 if σN(τ)≤τ, and sN+1(τ):=−1 otherwise. More precisely, we compute the limit points and the decay rate of the sequence (σN(τ)−τ)N∈N. Moreover, we give an accurate description of the behavior of the sequence of signs (sN(τ))N∈N, highlighting a surprising connection with the Thue–Morse sequence.
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